Properties

Label 2-552-24.11-c1-0-38
Degree $2$
Conductor $552$
Sign $0.879 - 0.475i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 0.915i)2-s + (1.49 + 0.881i)3-s + (0.324 + 1.97i)4-s + 1.83·5-s + (−0.800 − 2.31i)6-s + 4.30i·7-s + (1.45 − 2.42i)8-s + (1.44 + 2.62i)9-s + (−1.97 − 1.67i)10-s − 5.53i·11-s + (−1.25 + 3.22i)12-s − 1.97i·13-s + (3.93 − 4.63i)14-s + (2.72 + 1.61i)15-s + (−3.78 + 1.28i)16-s + 5.74i·17-s + ⋯
L(s)  = 1  + (−0.762 − 0.647i)2-s + (0.860 + 0.509i)3-s + (0.162 + 0.986i)4-s + 0.818·5-s + (−0.326 − 0.945i)6-s + 1.62i·7-s + (0.514 − 0.857i)8-s + (0.481 + 0.876i)9-s + (−0.624 − 0.529i)10-s − 1.66i·11-s + (−0.362 + 0.931i)12-s − 0.547i·13-s + (1.05 − 1.23i)14-s + (0.704 + 0.416i)15-s + (−0.947 + 0.320i)16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49880 + 0.379263i\)
\(L(\frac12)\) \(\approx\) \(1.49880 + 0.379263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.915i)T \)
3 \( 1 + (-1.49 - 0.881i)T \)
23 \( 1 - T \)
good5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 - 4.30iT - 7T^{2} \)
11 \( 1 + 5.53iT - 11T^{2} \)
13 \( 1 + 1.97iT - 13T^{2} \)
17 \( 1 - 5.74iT - 17T^{2} \)
19 \( 1 - 5.40T + 19T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 + 3.45iT - 31T^{2} \)
37 \( 1 - 6.42iT - 37T^{2} \)
41 \( 1 - 1.85iT - 41T^{2} \)
43 \( 1 - 8.60T + 43T^{2} \)
47 \( 1 + 2.85T + 47T^{2} \)
53 \( 1 + 2.24T + 53T^{2} \)
59 \( 1 - 0.271iT - 59T^{2} \)
61 \( 1 + 14.8iT - 61T^{2} \)
67 \( 1 + 3.34T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 3.52iT - 83T^{2} \)
89 \( 1 - 0.708iT - 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77241798583316529391764812077, −9.642186542932714332095937295781, −9.290244466120198408733161995476, −8.377178487831647697525293243399, −7.934358657802676046186492734208, −6.14605900604653547201721173815, −5.37415680992711829144718777188, −3.58649973586547202641280795106, −2.82977885307830635735823687360, −1.77382083660769930759276343586, 1.17426740881484636292113589879, 2.27749916217014478732451191508, 4.07217677155893567473495578190, 5.24712170093941355265085953152, 6.72264673234841804518217622706, 7.32964689345415319379322659545, 7.62223338320559755758200663579, 9.266764125459703335449867720617, 9.533981300085237123296832589384, 10.23891257149586245170466938020

Graph of the $Z$-function along the critical line